$L^p$-estimates for parabolic systems with unbounded coefficients coupled at zero and first order

TL;DR

This paper establishes $L^p$-estimates and conditions for the extension of evolution operators associated with nonautonomous parabolic coupled systems with unbounded coefficients, including gradient and $L^p$-$L^q$ estimates.

Contribution

It provides new sufficient conditions for the extension of evolution operators to $L^p$ spaces and derives related gradient and $L^p$-$L^q$ estimates for coupled parabolic systems.

Findings

Conditions for evolution operator extension to $L^p$ spaces.

Derived $L^p$-$L^q$ estimates for solutions.

Established gradient estimates in $L^p$.

Abstract

We consider a class of nonautonomous parabolic first-order coupled systems in the Lebesgue space $L^p({\mathbb R}^d;{\mathbb R}^m)$, $(d,m \ge 1)$ with $p\in [1,+\infty)$. Sufficient conditions for the associated evolution operator ${\bf G}(t,s)$ in $C_b({\mathbb R}^d;{\mathbb R}^m)$ to extend to a strongly continuous operator in $L^p({\mathbb R}^d;{\mathbb R}^m)$ are given. Some $L^p$-$L^q$ estimates are also established together with $L^p$ gradient estimates.